ESTATUTO TRIBUTARIO

Our computational study of manganese oxides, MnO, Mn3O4,α-Mn2O3, andβ-MnO2, us-

ing the rotationally invariant PBEsol+U+J approach, yields ground-state structural, mag- netic, and electronic properties of comparable quality and accuracy to previously reported hybrid functional and experimental studies. Our study shows that the limitations of con- ventional DFT regarding the magnetic and electronic structures of insulating transition metal oxides can be overcome mainly by pseudopotential design and careful selection of anisotropicU andJ values. The resulting noncollinear magnetic ground states [AFM-II, YK-FiM, NC-AFM2, and NC for MnO, Mn3O4,α-Mn2O3, andβ-MnO2, respectively] are

similar to the experimentally observed noncollinear magnetic configurations. All relaxed lattice constants, obtained with PBEsol alone, were in good agreement with the experimen-

tal values. Appropriate band gaps were obtained with U values smaller than those used

by previous GGA+U studies, while reproducing the electronic structure profiles in good agreement with those reported by previous hybrid functional studies. Our results over- all suggest the enhanced performance of our designed pseudopotential with semicore and partial core correction, thereby offering promising potential of the DFT+U+J approach for electronic structure studies involving other strongly correlated, complex noncollinear spin patterns and large-scale systems with accuracy nearing that of more computationally expensive methods such as hybrid functionals.

Chapter 4

Improper magnetic ferroelectricity of

nearly pure electronic nature in

cycloidal spiral CaMn₇O₁₂

Submitted toPhys. Rev. Lett.

4.1

Introduction

Multiferroics, simultaneously displaying ferroelectricity and intrinsic magnetic ordering, have gained much attention due to the complex physics underlying the magnetoelectric effect and its potential applications in spin-driven electronics, such as magnetic switch- ing of ferroelectric domains, voltage-controlled magnetic memory devices, four-state logic devices (up and down polarization and magnetization), and magnetoelectric sensors [152, 153]. The coupling strength between magnetism and ferroelectricity in a material highly depends on the origin of these two phenomena. In proper ferroelectrics, the polarization is a primary order parameter; it can be induced either by the ionic-covalent bonding be- tweend0transition metal ions and oxygen, as in BaTiO3, or by the stereochemically active

6s2 _{lone-pair, as in BiFeO}

3 [154, 155, 156]. Magnetism could only coexist with the latter

mechanism, since magnetism requires a partially filleddshell on the transition metal [152]. On the other hand, inimproperferroelectrics, nonzero polarization is stabilized by cou- pling to another order parameter. First, in electronic ferroelectrics [157], charge-ordering breaks inversion symmetry due to the formation of transition metal dimers of different va- lences and inequivalent bonds, as in (Pr,Ca)MnO3[158] and LuFe2O3[159]. Second, in ge-

ometric ferroelectrics, structural transition breaks inversion symmetry, as in YMnO3[160]

where geometric MnO5 tilting results in Y-O dipoles. Another example is hybrid improper

ferroelectrics such as Ca3Mn2O7 [161], where ferroelectricity is induced by antiferrodis-

tortive octahedral rotations of two nonpolar modes with different symmetries. Third, in magnetic ferroelectrics, spiral magnetic ordering breaks inversion symmetry, resulting in ionic and/or electronic displacements that give macroscopic polarization. Numerous exam- ples include: (a) orthorhombicRMnO3 (R= Tb, Dy, Tm) [162, 163, 164, 165, 166, 167],

CoCr2O4 [168], and MnWO4 [169] with cycloidal spiral, (b) triangular-lattice systems

with proper screw-type spiral [174], and (c) exchange-striction systems with collinear mag- netic structure, such as Ca3(CoMn)O6 [175], orthorhombicRMnO3 (R= Ho-Lu, Y) [176],

DyFeO3[177], and Ni3V2O8[178].

Materials with coexisting magnetism and ferroelectricity are classified into two groups, type-I and type-II [179], based on the nature of the order parameter coupling. Type-I con- sists of 6s2 lone-pair proper ferroelectrics and improper ferroelectrics of electronic and geometric origin including hybrid improper ferroelectrics, where ferroelectricity remains largely independent of magnetism. Type-II essentially refers to impropermagneticferro- electrics. Despite their relatively small spontaneous polarization and low Curie tempera- ture, type-II multiferroics are of great interest due to their stronger magnetoelectric effect. Type-II multiferroics with ferroelectric polarization emerging in response to the magnetic structure are of tremendous technological relevance, potentially leading to the design of robust room-temperature multiferroic materials with large spontaneous polarization and ultrafast switchability. In order to achieve this, theoretical insight into spin-induced polar- ization mechanisms is necessary.

Three microscopic mechanisms have been proposed to explain the emergence of spon- taneous ferroelectricityPin spin-spiral multiferroics [180, 181, 182]. First, the exchange striction model proposes that the symmetric exchange interaction in a ↑ ↑ ↓ ↓ spin order causes ferromagnetically coupled ions to move toward each other, generating P12 ∝ e12

(S1 ·S2) [182]. Here, P12 is the local polarization induced by the interaction between the

two spins on neighboring sites 1 and 2, S1 and S2 are the vector spins on the respective

sites, ande12 is a unit vector connecting the two magnetic ions. Second, two analytically

equivalent scenarios exist within the spin-current (KNB) model [183], where P12 ∝ e12

×(S1 ×S2): (a) In the inverse Dzyaloshinskii-Moriya (DM) mechanism, a nonmagnetic

anion moves in response to the DM interaction between the two canted spin sites [176]; (b) In the spin-current mechanism, the electronic charge distribution shifts in response to

the spin-current, defined asjs = S1 × S2 [183]. Third, the spin-dependentp-d hybridiza-

tion model arising from spin-orbit coupling (SOC) causes an intrasite polarization along the metal-ligand bond containing a non-zero spin component, generatingPml∝(Sm·eml)2

eml[171, 184, 185], whereemlis the metal-ligand unit vector.

Recently, CaMn7O12 manifested one of the largest magnetically induced ferroelectric

polarization measured to date (P= 2870µC/m2_{) [186]. Microscopic mechanisms involving}

the three models discussed above [182] have been proposed to explain the origin, magni- tude, and direction of the spin-triggered ferroelectricity in CaMn7O12: exchange striction

and DM interaction [187, 188], noncollinear exchange striction and spin-dependent p-d

hybridization [189], and inverse-DM mechanism [190]. However, none of these models provide a unified picture that explains the direction of the polarization, the charge density redistribution, and the role of ionic displacements.

In this chapter, we report on the ferroelectric polarization of nearly pure electronic na- ture in CaMn7O12 induced by its noncollinear cycloidal magnetic ground state, computed

via first principles density functional theory (DFT) calculations. For simplicity and clarity, we preserve inversion symmetry on the ionic lattice while the charge density distribution is permitted to respond to the symmetry-breaking spin pattern; these changes to orbital mixing make the dominant contribution to the polarization. Theoretically, we employ the generalized spin-current model [191] with a Heisenberg-exchange DM-interaction model to explain both the direction of the electronic polarization and the dependence of its mag- nitude on spin helicity.